Technion

Abstract:

It is well known that the thermocapillary effect can lead to the onset of two kinds of instabilities in a liquid layer heated from below: a short-wave instability generating hexagonal cells, and a long-wave instability connected with deformations of the fluid-fluid interface. In systems with one interface, the latter kind of instability is monotonic and leads to at the appearance of a dry spot or a stationary deformation of the interface. In systems with two interfaces, the coupling between interfacial deformations can produce a specific long-wave oscillatory instability which generates Maragoni waves with the unusual quadratic dispersion relation between the frequency and the wavenumber. The dependence of the critical temperature difference on the thicknesses of liquid layers can be non-monotonic.

Using the multiscale asymptotic expansion technique, we have derived the system of equations governing the long-wave deformations of the interfaces in the in the presence of the oscillatory Marangoni instability, assuming the bottom fluid layer to be very thin. The time evolution of the upper interface deformation is described by the Cahn-Hilliard equation with an additional term which provides the coupling with a linear equation for the lower interface. We investigate the bifurcation of traveling wave solutions and the interaction between the waves moving in different directions. By means of direct numerical simulations, we study the resonant phenomena and the transition to spatio-temporal chaos.